7. No work is done by gravity on a bowling ball resting or moving on a bowling alley because the force of gravity on the ball acts perpendicular to the surface. But on an incline, the force of gravity has a vector component parallel to the alley. Discuss the two ways this component accounts for (a) acceleration of the ball and (b) work done on the ball to change its kinetic energy.
In order to understand how the component of the force of gravity parallel to the surface of the incline accounts for acceleration and work done on the ball, we need to consider the concepts of force, acceleration, and work.
Force is a push or pull that can cause an object to accelerate or change its state of motion. Acceleration, on the other hand, is the rate at which an object changes its velocity. Finally, work is the transfer of energy that occurs when a force acts upon an object to move it.
(a) Acceleration:
When an object is placed on an incline, the force of gravity can be resolved into two components: one perpendicular to the incline and one parallel to the incline. The parallel component of the force of gravity is what accounts for the acceleration of the ball.
According to Newton's second law of motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this case, the parallel component of gravity acts as the net force, causing the ball to accelerate down the incline.
The equation for calculating the magnitude of the parallel component of gravity (force) is given by:
F = mg sinθ
Where F is the force, m is the mass of the ball, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of inclination.
So, the parallel component of gravity is responsible for the acceleration of the ball down the incline.
(b) Work Done:
Work is defined as the product of the force applied to an object and the distance over which the force is applied. In the case of the ball on the incline, the parallel component of gravity does work on the ball as it moves along the incline, changing its kinetic energy.
The formula for calculating work done on an object is:
W = Fd cosθ
Where W is the work done, F is the force, d is the displacement of the ball along the incline, and θ is the angle between the force and the displacement.
Since the angle between the force and displacement is 0 degrees on an incline, the equation simplifies to:
W = Fd
Therefore, the parallel component of the force of gravity does work on the ball by applying a force over a distance, resulting in a change in kinetic energy.
To recap, the parallel component of the force of gravity on the incline causes the acceleration of the ball down the incline and also does work on the ball to change its kinetic energy.